
theorem
  for T being non empty LinearTopSpace,
      L being non empty transitive reflexive RelStr,
      f being Function of [#]L,the carrier of T,
      x being Point of T,
      V being local_base of T st [#]L is directed holds
  x in lim_f f
    iff
  (for v being Subset of T st v in (V/\NeighborhoodSystem 0.T)
  ex i being Element of L st for j being Element of L st i <=j holds
    f.j in x+v)
  proof
    let T be non empty LinearTopSpace,
    L be non empty transitive reflexive RelStr,
    f be Function of [#]L,the carrier of T,
    x be Point of T, V be local_base of T;
    assume
A1: [#]L is directed;
    set B={x+U where U is Subset of T:U in V &
    U is a_neighborhood of 0.T};
    reconsider B as Subset-Family of T by Lm2;
    now
      hereby
        assume
A2:     x in lim_f f;
        let v be Subset of T;
        assume v in (V/\NeighborhoodSystem 0.T);
        then v in V & v in NeighborhoodSystem 0.T by XBOOLE_0: def 4;
        then v in V & v is a_neighborhood of 0.T by YELLOW19:2;
        then x+v in B;
        then reconsider b=x+v as Element of B;
        consider i0 be Element of L such that
A3:     for j be Element of L st i0 <= j holds f.j in b
        by A1,A2,Lm3;
        take i0;
        thus for j be Element of L st i0 <=j holds f.j in x+v by A3;
      end;
      assume
A4:   for v be Subset of T st v in (V/\NeighborhoodSystem 0.T)
      ex i be Element of L st for j be Element of L st i <=j holds f.j in x+v;
      for b be Element of B ex i be Element of L st
      for j be Element of L st i <=j holds f.j in b
      proof
        let b be Element of B;
        B is non empty by Lm2;
        then b in B;
        then consider U1 be Subset of T such that
A5:     b=x+U1 and
A6:     U1 in V and
A7:     U1 is a_neighborhood of 0.T;
        U1 in NeighborhoodSystem 0.T by A7,YELLOW19:2;
        then U1 in (V/\NeighborhoodSystem 0.T) by A6,XBOOLE_0:def 4;
        then consider i0 be Element of L such that
A8:     for j be Element of L st i0 <= j holds f.j in x+U1 by A4;
        take i0;
        thus thesis by A5,A8;
      end;
      hence x in lim_f f by A1,Lm3;
    end;
    hence thesis;
  end;
